Elements of the Theory of Functions of a Complex Variable

ELEMENTS OP THE

THEORY OF FUNCTIONS COMPLEX VARIABLE WITH

ESPECIAL REFERENCE TO THE METHODS OF RIEMANN

BY

DR. H.

DUREGE

LATE PROFESSOR IN THE UNIVERSITY OF PBAGTJE

AUTHORIZED TRANSLATION FROM THE FOURTH GEJtMAtf EDITION' BY

GEORGE EGBERT FISHER,

M.A., PH.D.

ASSISTANT PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF PENNSYLVANIA

AND

ISAAC

J.

SCHWATT,

PH.D.

INSTRUCTOR IN MATHEMATICS IN THE UNIVERSITY OF PENNSYLVANIA

PHILADELPHIA G.

E.

F1SHEB AND 1896

I.

J.

SCHWATT

OOPTKIGHT, 1896, BY .

E.

PISHES AND

HorSuooti 3. S.

I. J.

SCHWATT.

lwi$

Cushmg & Co. Berwick & Smith Norwood Mass. U.S A.

TRANSLATORS' NOTE.

WE

desire to express our indebtedness to Professor

McMahon, translation, J. S.

Cornell University, for

and

Gushing

Ms

kindness in reading the

for his valuable suggestions.

&

Co.,

James

Also to Messrs.

Norwood, Mass., for the typographical

excellence of the book.

G. E. F. I.

UNIVERSITY OF PENNSYLVANIA, PHILADELPHIA, December, 1895.

J.

S.

FROM THE AUTHOR'S PREFACE TO THE THIRD EDITION. NUMEROUS

additions have been

made

treats of multiply connected surfaces.

to Section IX., which, If Riemann's funda-

mental proposition on these surfaces be enunciated in such a form that merely simply connected pieces are formed by both modes of resolution, as is ordinarily, and was also in then the it be supplemented for further must 49, case, 52 Such supplementary matter was given in applications. in the classification of surfaces, and in But if we 53, V. express the fundamental proposition in the form in which

Riemann originally established it, in which merely simply connected pieces are formed by only one mode of resolution, while the pieces resulting from the other mode of resolution may or may not be simply connected, then all difficulties are obviated, and the conclusions follow immediately without requiring further expedients. This was shown in a supplementary note at the end of the book.

AUTHOR'S PREFACE TO THE FOURTH EDITION. IN the present new edition only slight changes are made, consisting of brief additions, more numerous examples, different modes of expression, and the like. In reference to the above extract from the preface of the preceding edition, I have asked myself the question, whether I should not from the beginning adopt the original Riemann enunciation of the fundamental proposition instead of that which is given in 49. Nevertheless, I have finally adhered to the previous arrangement, because I think that in this way the difference between the two enunciations is made more prominent, and the advantages of the Riemann enunciation are

more

distinctly emphasized.

H. PRAGUE,

April, 1893.

DUREGE.

CONTENTS. PAGE

INTRODUCTION

1

SECTION

I.

GEOMETRIC REPRESENTATION OF IMAGINARY QUANTITIES. 1.

A complex quantity x + ty or r(cos

-4-

i

sin -

Q

THEORY OF FUNCTIONS.

42 in which

a, b, c

denote three complex constants, and therefore

=

three points, z a is a branch-point; at which all three values b all c. of the function become equal to w Moreover, at z three values of w become infinite. The three functions suffer

=

=

here

art

interruption of continuity, and hence it can remain is to continue its course,

undetermined on which path each

because, when the function makes a spring, it can just as well spring over to the one as to the other continuation of its path.

=

b is likewise a branch-point. Also, as a general w or discontinuous which infinite at becomes those rule, points are branch-points. Exceptions to this, however, may occur;

Therefore

there are cases in which points are not branch-points, although at them the values of the functions are either equal or infinite, This, for the present, can only be illustrated by examples. the functions

v _ -V/-1 J.

z

=+1

and

z

=

y2.

%r\C\ of increasing in the of circle direction the the circumference increases and to constant and r remains 1, equal angles, If therefore the variable return to the point to 2 TT. from

8

= 1,

then

= cos 2 + i sin 2 w = v# = cos + sin = z

and therefore

TT

TT

thus the function does not original value

+ 1,

TT

now have

TT?

1;

at the point z

but acquires the other value

1.

=1

the

The very

THEOET OF

48

place, when the variable describes any other closed round the origin starting from z = 1 for this path can be deformed into the circle by gradual changes without thereby passing over the origin. In general, if w start with the value W Q from any point z for which

same takes line once

;

,

ZG

= TQ (cos

W = r ^ (cos

and therefore

+




at

w2w

5,

W$W G each by ,

interchange their values, without a value from one pair

ever entering another. If we let z describe a closed line including two branch-points, we can again replace such a one bv two successive circuits,

each round one point. If the points a and b be enclosed, we We have the same condition as in the preceding example. will therefore follow only circuits round a and c, and tabulate the results below.

Therein

w

tive circuits

acquires its original value only after six consecu-

round the points a and

c.

11. The preceding considerations show that, given a multiform function, we can pass continuously from one of the values

THEORY OF FUNCTIONS.

56

which, the function can assume for the

same value of the

vari-

able to another, by assigning complex values to the variable and letting it pass through a series of continuously successive values,

which ends with the same value with which

it

began

(geometrically expressed, by letting the variable describe a closed line). It has been further shown that a definite and

continuous series of values of the variable (a definite path), also, always leads to a definite function-value, except in the single case when the path of the variable leads through a branch-point, a case, however, which can always be obviated by letting the variable make an indefinitely small deviation in

the vicinity of the branch-point. 1 This naturally suggests the desirability of avoiding the multiplicity of values of a multi-

form function, in order to be able to treat such a function as if it were uniform. According to the preceding explanations, it is necessary for this purpose only to do away with the multiplicity of paths which the variable can describe between two given Now Cauchy has already remarked that this could be points. effected, at least to

a limited extent, by demarcating certain which the variable z is supposed to be

portions of the plane in

moving and not permitting the

latter to cross the

boundary of

Tor, since a function, starting from a point # of the variable, can assume different values at another point z ly only when two paths described by the variable enclose a

such a region.

branch-point ( 9), it is always easy to mark off a portion of the 2-plane within which two such paths from ZQ to z1 are not

by drawing certain lines which start from branchand which are not to be crossed, to make such paths Within such a region the function remains uniimpossible.

possible, or,

points,

^ only a single value along then called monodromic (after paths. or one-valued uniform, Cauchy) (after Riemann). Although

form, since

it

this

acquires at each point

The function

all

method

is

is

of great advantage, for instance, in the evaluar

If we regard the position of the path of the variable, in case it lead through a "branch-point, as the limiting position of a path not meeting the branch-point, then to this assumption corresponds again a definite func1

tion-value.

MULTIFORM FUNCTIONS.

57

tion of definite integrals, nevertheless by means of it only a definite region of values, or, as Riemann calls it, a definite

branch of the multiform function, is separated from the rest In order to be able to treat an algeitself.

and considered by

biaic function in its entirety

and yet

as if it were a uniform

function, Eiemann has devised another method, -which will be set forth in the following.

Riemann assumes that, when a function is ^-valued, when therefore to every value of the variable n values of the function correspond, the plane of z consists of n sheets or leaves (or that n such sheets are extended over the 0-plane), which together form the region for the variable. To each point in each sheet corresponds only a single value of the function, and to the n points lying one immediately below another in all the n sheets correspond the n different values of the function which belong Now at the branch-points, where sevto the same value of z. eral f anction-values, elsewhere different, become equal to one another, several of those sheets are connected, so that the particular branch-point is supposed to lie at the same time in all these connected sheets. The number of these sheets thus con-

nected at a branch-point can be different at each branch-point, and is equal to the number of function values which change one into another in cyclical order for a circuit of the variable

round the branch-point. In the paragraph, wherein the function

example of the preceding six-valued, we shall assume

last is

the 2-plane as consisting of six sheets. branch-points a and b the values w ly w%

E/ound each of the on the one hand,

ws

,

and MI, w59 WQ on the other, change one into another hence we assume that at each of these points the sheets 1, 2, 3, on the one hand, and the sheets 4, 5, 6, on the other, are connected. Bound the point c, however, firstly, Wi and w4 secondly, w2 and w5 thirdly, and W Q change respectively one into the other hence at the point c, first the sheets 1 and 4, then the sheets 2 and 5, and finally the sheets 3 and 6 are connected. Now ;

,

;

j

%

,

;

for the purpose of exhibiting the continuous change of one 1 value of the function into another, so-called branch-cuts are 1

Sometimes called

cross-lines.

(Tr.)

THEORY OF FUNCTIONS.

58 introduced.

These are quite arbitrary lines (except that one from a branch-point itself), which either pass or join with each other two branch-points. We do

cannot intersect to infinity,

not suppose the sheets to be connected along these branch-cuts as they naturally lie one above another, but as the function-

round the respective branch-points. If, for in the last instance, example of the preceding paragraph, we draw a branch-cut from a to b (Fig. 14), we then, in making a

Tallies interchange

round the point a in the direction of increasing angles, connect the sheet 1 with the sheet 2 along the branch-cut, then 2 with 3, and finally 3 with 1 again. Let us call the right side circuit

of the branch-cut ab, that which an observer has on his right, when he stands at a and looks toward "b. Then if z go from a

point #

in sheet 1 (w with the value

w$ and make

a circuit

FIG. 14.

round the point a in the direction, of increasing angles, on crossing the branch-cut from right to left, it passes from the first

sheet into the second

returns to #

>

and

is still

r rather to the point

in the latter

when

it

g lying immediately below

w acquires the value w2 If the description, be still continued, z passes, after crossing the branch-cut a second time from right to left, into the third sheet

#

in the second sheet, so that

and

is

still

in the

same when

ated in this sheet below z

;

w

arrives at the point

it

has

now

7i

situ-

acquired the value

w

,.

Finally, when z crosses the branch-cut a third time, we assume that the right side of the third sheet is connected along the

branch-cut with the left side of the first sheet through the second sheet, so that z crosses from the third sheet to the first sheet and then returns again to # .

MULTIFOEM FUNCTIONS.

59

Not until now is the line actually closed, and has w acquired again its original value. In Fig. 14 the lines are denoted by the numbers of the sheets in which they run, and, in addition, those in the second and third sheets are thickly dotted and thinly dotted respectively. The points Z Q, g, h, which ought really to be one directly below another, are, for the sake of

drawn

side

by side. must be imagined in the case of all branchpoints, and since from each such point a branch-cut starts, the variable cannot make a circuit round the branch-point without crossing the branch-cut, and thereby passing in succession into all those sheets which are connected at the branch-point. How the branch-cut is to be drawn in each case depends upon the function to be investigated, and can generally be chosen in different ways. In our example a and b may be connected by such a cut, because with circuits round the point clearness,

A similar course

& in the direction of increasing angles the function w-^ changes into WQ and this into w 2 (Fig. 14), and hence at b the same sheets are connected as at a, and in the same way ; namely,

the right side of 1 with the left of 2, the right side of 2 with the left of 3, and the right side of 3 with the left of I. 1

Let us continue with this example and investigate the

cir-

If we wish to exhibit this method of representation by a model, a difficulty arises, first, because the sheets of the surface interpenetrate, and in the second place, because frequently at branch-points several sheets, which do not lie one immediately below another, must be supposed to 1

But for the purpose of illustration, it is for the most part necessary only to be able to follow certain lines in their course through the different sheets of the surface. This can be easily effected in the following way Pirst cut in the sheets of paper placed one above another, be connected.

:

which are to represent the surface, the branch-cuts, and then only at those places where a line is to pass over a branch-cut from one sheet into another join the respective sheets by pasting on strips of paper. Then we can always so contrive that, when the line is to return to the first sheet, from which it started, we have the necessary space left for the fastening of the strip of paper by means of which the return passage is effected. By these attached paper strips union of the separate sheets into one connected surface is accomplished; and it is then no longer necessary to connect the sheets with one another at the branch-points.

THEORY OF FUNCTIONS. cults discussed in the preceding paragraph round a and b, and round a and c. For a circuit round a and 6 the branch-cut is

not crossed at

fact

w

(Of.

Ex.

and

e,

resumes 3,

let

10).

all,

so that z remains in the first sheet; in

value at

its original

To examine the

us draw from

the sheets of every pair each other.

c to infinity I,

4

;

2,

5

;

#

after such a circuit.

round the points a a branch-cut, and let here

circuit

3,

6 pass respectively into

For the passages of the function-values taking place here,

we have found

the following table

(p.

55)

:

These passages are represented in Fig. 15, by designating each line by the number of the sheet in which it runs. The points properly lying below the initial point 1 are represented

MULTIFORM FUNCTIONS.

61

side, for the sake of clearness, and the last point 1 is supposed to coincide with the first. This region for the variable #, consisting in our example of six sheets, forms a single connected surface, since the sheets are connected at the branch-points and pass into one another along the branch-cuts. In this surface w is a perfectly uniform function of the position in the surface, since it acquires the same value at every point of the latter, along whatever

side

by

to be

If z describe path the variable may arrive at the point. between two points two paths, which enclose a branch-point, then one of the two must necessarily cross a branch-cut and

thereby pass into another sheet, so that the terminal points of the two paths are no longer to be considered coincident, but as two different points of the ^-surface, at which different

But if z describe an actually closed curve, i.e., if the initial and the terminal points of the curve coincide in the same point of the same sheet, then also the function acquires its initial value. Only when the variable function-values occur.

passes through a branch-point can it pass at will into any one of the sheets connected at that point, and in that case it remains undetermined which value the function assumes ( 8). 12. Now in order to prove that we can also, in general, transform an ^-valued function into a one-valued by means of

an 7i-fold surface covering the #~plane, the single sheets of which are connected at the branch-points and along the branchcuts in the manner explained above, we first assume the 2-plane to be still single, and let the variable z, starting from an arbitrary point zQ describe a closed line, which makes a circuit once round a single branch-point and does not pass through any other branch-point. At 2 the function possesses n values let us assume them to be written down in a certain sequence. IsTow after the variable has described the closed line and returned again to # each of the above n function-values will have either passed into another or remained unchanged. Since j

;

,

is again at the point z0} these new values of the function cannot differ from the former in their totality; but

the variable z

THEORY OF FUNCTIONS.

62 if

we suppose them

to be written

down

in the sequence in

which they have arisen in succession from the former, they will occur in an arrangement different from the previous one. Any one arrangement of n elements, however, can be derived from another arrangement by a series of cyclical permutations. By a cyclical permutation of the pth order we understand one in which from the existing n elements we take out p arbitrarily and in the place of the first of these put a second, in the place of the second a third, etc., and finally in the place of the ptli the first. Such a cyclical permutation of the pth order has the property, that after p repetitions of it, and not sooner, the arrangement is restored for, since the place of each is taken by another, that of the pth element, however, by the first, each element can reappear in its original place 1 other elements have occupied the same only after all the p place; then, however, each element actually returns to its Now, to prove that each arrangement can be original place. derived from another by a series of cyclical permutations, we assume that any one arrangement arises from another by subThe place of stituting one element, say 3, for another, say 1. 3 is then taken either by 1, in which case we have already a cyclical permutation of the second order, or by some other element, say 5. The place of the latter is then again taken either by the first, 1, in which case we have a cyclical permutation of the third order, or again by another which is different from those already employed, 1, 3, 5. Its place can be taken either by the first, whereby a cyclical permutation would be closed, or again original

;

element

finally, however, the cyclical permutation must because terminate, altogether there is only a finite number of elements, and the first element 1 must be found in some place

by another;

of the second arrangement. In. this way then a series of eleis disposed of. If we next begin with some one of the

ments

we can repeat the former procedure have been exhausted, and we thus obtain

elements not yet employed, until all the elements

a certain number of cyclical permutations which, employed either successively or also simultaneously, produce the second

arrangement from the

first.

If an element have not changed

MULTIFORM FUNCTIONS.

63

its place in the second arrangement, such a permanence can be regarded as a cyclical permutation of the first order. Let us illustrate the above by an example. Suppose the elements

123456789

10

11

have passed into the arrangement

it is

31152710196845 row 1357

evident that the

3571;

has changed into

these therefore form a cyclical permutation of the fourth If we next proceed from 2, it is evident that the row

order.

2

42;

11

changes into

4

11

we have a second cyclical permutation, of the third The next element not yet employed is 6. Then the

therefore order.

row changes into

6

10

8

9

10

8

9

6,

and we have a third

cyclical permutation, of the fourth order. 11 elements are exhausted, and consequently the second given arrangement is derived from the first by the three cyclical permutations obtained above.

Now

all

If we now return to our function-values, it follows that, whatever arrangement of them may have arisen from the circuit of the variable round the branch-point, it can be produced

by a

series of cyclical

the variable be

permutations of the function-values. If to describe a circuit round the same

made

branch-point a second time, each function-value undergoes the same change that it experienced the first time. For this second circuit, therefore, the cycles remain the same as for the first,

and

so, too,

for each subsequent circuit.

Thus the values

THEORY OF FUNCTIONS.

64

of the function (unless they all form a single cycle, which case can also occur cf. Ex. 3, 10) are divided at each branch-point into a series of cycles, so that in each cycle only certain ;

definite values of the function

with the total exclusion of cycle (cf. Ex. 4, 10).

can permute among themselves values contained in another

all

If a single value of the function do not

change for the cirround the branch-point, the same can be has been remarked above, as forming by itself

cuit of the variable

regarded, as a cycle of the

first order.

If the variable be

now made

to

some quite arbitrary closed line, the latter can be deformed into a series of circuits round single branch-points describe

(

9).

Therefore the arrangement arising through this closed produced by the cyclical permutations occur-

line can also be

ring at the branch-points. If, now, the ^-surface be supposed to consist of n sheets, the preceding justifies the assumption that, at each branch-point, certain sets of sheets are supposed to be connected, which continue into one another along the branch-cuts in the way above Then the variable, by describing a circuit round a stated. branch-point, passes in succession into all those sheets belong-

ing to the same group, and into none but those, and finally returns into that sheet from which it started.

it

To find for an ^-valued algebraic function the Eiemann n-sheeted surface, let us first determine its branch-points and choose some definite value # of z, which, however, must not itself be a branch-point. must then let z describe a cir-

We

round each separate branchfrom z and returning again to it, and point, always starting we must ascertain how the function-values occurring at # cuit in the single 0-plane once

are divided at each branch-point into the cycles,

and how, within

above-mentioned with one another. 1 these, they permute

" Puiseux, Recherches sur les fonctions alg6briqu.es," T. xv. de Math. Journ. Liouville, (In the German by H. Fischer, Puiseux s Untersuchungen uber die algebraischen Funktionen. Halle, 1861.) Konigsberger, Vorlesungen uber die Theorie der ell. Funktionen. Leipzig, 1874. 1

On this point cf.

1

I.

S. 181.

MULTIFORM FUNCTIONS.

65

This having been ascertained, if in the ^-sheeted surface the points of the w-sheets which represent the value # be desigzn , so that the subscript nated in succession by %, 22 , ? indicates the sheet in

can ZQ

which the point

first arbitrarily distribute

among the sheets; way which of

definite

is

situated, then

we

the values of the function at

i.e., we can assume in an arbitrary but these values of the function shall belong

to each of the points

%, z>

,

#M

.

We

will denote these

w2 , -, io n Let us next draw from values in order by each branch-point a branch-cut to infinity; and for each of the

w,

latter let us so

.

determine the connection of the sheets that

it

shall exactly correspond to the cycles previously ascertained. If, therefore, in the single 2-plane, for a single circuit round a Q

W K into W A etc., certain branch-point, w a be changed into then in the ^-sheeted surface the connection of the sheets is

w,

,

round the same branchz^ from % K to # A etc. If a single function-value w^ suffer no change thereby, the corresponding sheet ^ is connected with no other sheet, so so determined that for a single circuit point, the variable passes from z a to

,

that the sheet p, continues into itself along the branch-cut, it being therefore unnecessary to draw the branch-cut in this sheet. If, as has been assumed, a branch-cut extend from each branch-point to infinity, the connection of the sheets can be determined at each branch-point independently of the others.

This, however, does not exclude the possibility of sometimes connecting two branch-points by a branch-cut, or of making

from one branch-point; but this occur when the only may previously ascertained way in which the function-values permute at the respective branch-points permits such an arrangement. Thus, in the example of a sixseveral branch-cuts extend

valued function considered in the preceding paragraph, a branchcut can be drawn from each of the three branch-points a, &, c

way in which the values of the function round a and b also admits of a and b being connected permute a branch-cut. by These determinations having once been established, the to infinity; but the

THEORY OF FUNCTIONS.

66

among the

function-values for each, value of z are distributed

To prove

n

sheets in a definite way. 2~plane the different values

the single

this, since in

which the function can have

at

one

and the same produced only by the different paths to 2, it is only necessary to show that, if in the n-sheeted surface starting from a certain point, say %i, and with the definite point z are

value

%,

any two

we reach the same arbitrarily chosen point ZK along we are always led to the same function-

different paths,

value (by the two paths). In this proof different cases must be distinguished. the terminal point of a path (1) Let us first assume that starting from z* is one of the points representing the value # Then the corresponding path in the single #-plane say #x ?

.

forms a closed

line.

This can be deformed into a series of

(closed) circuits round single branch-points without changing the final value of the function. Corresponding to this in the

makes circuits round single after each circuit goes to whichever of the 3n it can reach. The points at which it arrives

n-sheeted surface, the variable also

branch-points and

points z, 2 2 , -, in this manner may be designated in order by ^, # a z, *, Z K 2. According to the principles established above concern,

ing the arrangement of the function-values, and since here only single circuits round any one of the branch-points are taken into consideration, it follows that w assumes in succession the values W], w a , Wp, , w^. 3STow a similar deformation can be made in the case of any other path starting from z and

w^

terminating in ^, although then the variable will generally pass into other sheets than before. Let us assume that the

from z to za zb *, #/, then the corresponding series of functionwk and since the last circuit, accordw^,

circuits lead the variable in succession

and

finally to #A

,

,

,

is Wi, w a , ing to the assumption, leads from z to #A , Wjf must finally also This may be illustrated by an example. change into W A

values

,

,

.

In the six-sheeted surface represented in 11 we can reach for instance ZB from z by two circuits round the point a, by which we come first from z to 2 and then to 23 But among others we can also choose the following way from % round .

,

:

MULTIFORM FUNCTIONS.

67

a to %, then round c to %, then round a to z6 and finally For the first path w? assumes the values w, w 2 round c to #8 w3 for the second, %, w2 %, w6 w 3 but the final value is the same for both paths. It must be emphasized as a special ,

.

,

,

,

;

,

case that, if the variable return by any one path to its startingpoint zf, the function also resumes the original value wf.

(2) We next consider two paths A and B connecting the same points and running entirely in one and the same sheet therefore, if for instance, we assume again % as the startingThen, if the two paths enclose point, entirely in the sheet 1. no branch-point, no special discussion is required, since the cor;

responding paths in the single #-plane also enclose no branchpoint, and therefore lead to the same value of the function

be noted that this case can occur, if the paths branch-points, only when from neither of the enclosed branch-points a branch-cut extends to infinity for, otherwise, at least one of the paths would have to cross the branch-cut and could not run entirely in the same sheet. The case under consideration can therefore only occur when (

9).

It is to

considered enclose

;

several branch-points are enclosed by the two paths, and when each branch-cut connects two branch-points with each other. If then, in the single 2-plane, we let circuits round all the enclosed

branch-points precede one of the paths, say J3, we obtain a new path C leading to the same function-value as does A. But if

made in the n-sheeted surface, they again lead back to 0!, because each branch-cut connects two branch-points, and therefore, for the successive circuits round the latter, the branch-cut must be crossed twice in opposite directions; we thus always come back to the sheet 1, and therefore finally also to 0JL Then, according to what has been proved in the these circuits be

.

preceding case, the function also acquires again at #! the value Wi. Since, however, the path (7, which consists of the circuits and the path J5, leads to the same value as does A, and since

B

the function starts on the path with the value w, therefore must also lead to the same value as does A.

this alone (3)

to be

Finally, let us assume as the terminal point of the paths examined any point #A lying in any arbitrary sheet A. Let

THEORY OF

68

FUNCTION'S.

the initial point, as before, be %. For such a path we can, in the first place, without changing the final function-value, sub-

and then, running the corresponding paths in the single #-plane ; the terminal portion of the second can always "be so chosen that this can be deformed into the first without stitute another path,

which

entirely in the sheet A, to z k

first

;

leads to 2 X

for, in

If we now make the necessarily crossing a branch-point. same deformation in the two different paths, both first lead to

here the function according to (1) acquires along both The portions of the two paths still remainpaths the value W A #A

;

.

ing run entirely in the sheet X, commence at the same point 2A and with the same value of the function wx therefore, both ;

according to (2) also lead to the same value of the function at z k "We have thus proved that after the above determinations, .

chosen quite arbitrarily, have been made, the function acquires at each point of the surface a definite value independent of the path and becomes a one-valued function of the position in the surface. Thereby we have removed the multiformity of algebraic functions, and in what follows we shall now always assume that the region of the variable consists of as many sheets as are required to change the multiform function under 1

consideration into a uniform one, and

points as identical, only sheet of the surface.

when they

we

two same

shall consider

also belong to the

Accordingly we shall call a line actually closed, only when and terminal points coincide at the same point of the same sheet. On the other hand, should a line end in a point

its initial

situated in another sheet above or below the initial point,

we

shall sometimes call such a line apparently closed.

13.

To the above

considerations let us add a few remarks.

In crossing a branch-cut, one sheet 1

Though we can

is

continued into another,

also ascribe branch-points to such functions as log 0, should then be obliged to assume that an infinite number of sheets of the surface are connected in a branch-point. For this reason the functions mentioned will be considered later from the point of view

tan- 1 ^,

etc.,

we

of functions defined

by

integrals.

MULTIFORM FUNCTIONS.

69

set forth above, in such, a manner thai, when the moves in it, the function changes continuously. It follows from this, which is to be well heeded, that the function in the same sheet must always have different values on-the two

been

as lias

variable

Let us assume, for example, that a sheet a beyond branch-cut into another sheet X, and be two points representing the same value of 2,

sides of a branch-cut. K is continued

and 2A and lying in

let Z K

K and X respectively and infinitely near the branchFurther, let z' K be the point which lies in K on the other side of the branch-cut directly opposite % K and infinitely near

cut.

the same.

must

first

Then the variable, in order to pass from ZK move round the branch-point to z K} from which

to #A?

r

point

immediately comes to #A by crossing the branch-cut. Accordingly z K and ZK are in continuous succession Z K and # A however,

it

f

j

If

are not.

WK

,

w' K

,

,

w k denote the function-values

corresponding continuous with w' K9 but not

then w^ is K z'K #x respectively, with w K9 and if we disregard the infinitely small difference between wx and w'K) we can say w'K = W K but since wx is different from W K W K and w'K are also different from each other. Take, for instance, the function w = V#. The surface then consists of two sheets, which are connected at the branch-point s = 0. Here wx = WK (cf. 10, Ex. 1) therefore w K = WK to

j

,

]

,

r

;

.

In this example, therefore, the values of the function in the same sheet have opposite signs on the opposite sides of the branch-cut.

Eiemann calls the branch-points also winding-points, because the surface winds round such a point like a screw surface of Then, if only two sheets of the surface infinitely near threads. be connected at such a point, it is called a simple branch-point, or a winding-point of the first order; if, however, n sheets of the surface be connected at it, it is called a branchpoint or E"ow, for many invesT)th order. tigations, it is important to show that a winding-point of the 1 simple order may be regarded as one at which n (n winding-point of the (n

l)th If we assume, for example, branch-points have coincided. n 5, then at a branch-point in which 5 sheets are connected,

=

the variable passes after each circuit into the next following

THEOBY OF FUNCTIONS.

70 sheet,

and a curve must make 5

circuits

round a branch-point

arrives again in the first and becomes closed. But the this property such a point is characterized.

before

By

it

same

we assume 4 simple branch-points a, 6, c, d which the following sheets are connected in succession:

also takes place if

in at

a

land

2

d

b

c

1 and 3

1 and 4

1 and

5.

In Fig. 16, aa'y W, cc , dd are the branch-cuts, and the figures refer to the numbers of the sheets in which the lines run. f

f

If the curve, starting from # (Fig. 16), cross the section aa', passes from 1 into 2 and remains in 2 for the entire circuit,

it

because this sheet

is

not

co>n-

nected with another at any of the points &, c, d. Thus, for the first circuit the curve passes

from 1 into 2. If aa be crossed a second time, the curve passes from 2 into 1, and then at from 1 into 3. Then, however, l

W

it

remains in 3 until

%

to

fore carrying

W

its

return

the second circuit thereit

into 3.

Only at

again passes from 3 into and then at cc from 1 into it

f

1, 4.

In

this

way

each

new

cir-

cuit carries the curve into the next following sheet therefore, after the fifth circuit the curve returns into the first sheet and ;

becomes

closed.

here in the same fourth order.

It is thus seen that the passages take place way as in the case of a winding-point of the

Therefore, by

making the four simple branchapproach one another and

points, as well as the branch-cuts, finally coincide,

we

obtain a branch-point of the fourth order.

This simple example shows at the same time that the number of circuits which a curve must make round a region in order to become closed, exceeds by 1 the number of the simple branchpoints contained in this region, since the winding-point of the

MULTIFORM FUNCTIONS. fourth order will be

is

shown

71

equivalent to four simple branch-points.

It

later that this relation holds generally.

14. For the complete treatment of the algebraic functions requisite for us to take into consideration infinitely

it is still

In the plane in which z is large values of the variable z. can move from variable this moving, any given point, e.g., from the origin, in any direction to infinity. But if we now suppose the plane to be closed at infinity, like a sphere with an infinitely large radius, we can imagine that all those directions

extending to infinity meet again in a definite point of the and accordingly the value z oo can then be represented by a definite point on the spherical surface. The same

=

sphere,

representation may be obtained by conceiving that the #~plane tangent at the origin to a sphere of arbitrary radius. Let us suppose the point of tangency to be the north pole of the is

Then any point z of the plane can be projected on sphere. the surface of the sphere by drawing a straight line from the south pole s of the sphere to the point z, and cutting with this line the surface of the sphere. But if the infinitely distant points of the 3-plane be projected in this manner on the surface

of the sphere, the projections all fall in the point s, by which oo is represented in that case. point therefore the value z If, now, the 0-plane consist of %-sheets, the spherical surface

=

can be supposed also to consist of

7i-sheets,

and we can assume

=

oo lie that the points of the ?i-sheets representing the value z that it one above another. Then is also conceivable directly

several sheets are connected at the point oo

a branch-point. Given a function whether z = oo is a branch-point, is

z

=

If,

then, f(z) change into

of /(#) furnishes for

each branch-point u for f(z)

;

therefore

according as

u=




= f/K +

*'

/*o

and

this equation holds whatever may be the functions $ and determining the path of integration. By reducing in it / also to the form of a complex quantity, we axe led to only real

\f/

integrals,

and hence we can apply to the former the rules

of differentiation holding for real integrals.

We

therefore

obtain

Hence

^

Sy

=

&' ;

It then follows, 5) w is a function of z. of the propositions stated above, that w can also be considered as a function of the lower limit if the

consequently (by

from the second

upper one be regarded as constant.

it

dw

Bw

dz

=/(. w

follows also that

On that,

Since,

further

(

5),

the other hand, the proposition holding for real integrals, denotes a function of #the derivative of which

when F(z)

is /(*),

1

This result follows also from the sum (1), by which the integral is if we separate in it the complex quantities into their constituent

defined, parts.

THEORY OF FUNCTIONS.

84

cannot without further limitation be applied to complex insuch integrals, as has already been remarked, depend not only upon the upper and lower limits, but also upon the whole series of intermediate values,

tegrals, because the values of

i.e.,

upon the path

of integration.

17. In order to examine the influence of the path of integrar upon the value of the integral, we shall commence with

tion

the following considerations.

Let

be the variable, accordingly x and y the rectangular co-ordinates of the representing point. If we have a region of the plane definitely bounded in some way, which can consist of either one or several sheets, and if P and Q be two real functions of x and y which for all points within the region are finite and continuous, then the surface integral

extended over the whole area of the region,

is

equal

to the linear

integral

taken round the whole boundary of the region.

We case,

shall not only prove this proposition for the simplest the region consists of only one sheet and is bounded

when

we

same time take which the boundary consists of several separate closed lines, which can either lie entirely outside of one another, or of which one or more can

by a simple closed

line,

but

shall at the

into consideration those cases also in

be entirely enclosed by another. Finally, we shall not exclude the case when the region consists of several sheets which are connected with one another along the branch-cuts. Yet we shall then

at

assume that the region does not contain any branchwhich the functions P and Q become infinite or

points discontinuous.

It

is,

however, necessary, in order to include

INTEGRALS WITH COMPLEX VARIABLES. all

those eases, to determine more definitely

boundary-direction. positive directions

If

we assume,

of the oo

and

tlie

85

meaning of

as is customary, that the the ?/-axis lie so that an

observer stationed at the origin and looking in the positive direction of the aj-axis has the positive y-axis on his left, then let

us so assume the positive boundary-direction that one who it in this direction shall always have the bounded area

traces

of the region

on his

At each point

The same can be expressed thus boundary the normal, drawn into the situated with reference to the positive

left.

of the

:

interior of the area, is direction of the boundary just as is the positive ^-axis with refer-

ence to the positive #-axis. If, for instance, the boundary consist of

an external closed line and a circle lying wholly within the same, so that the points within the circle are external to

the bounded

area of the region, then on the outer line the positive boundaryis that of increasing anwhile on the inner circle it is

direction gles,

the opposite, as is shown by the arrows in Fig. 21. ISTow in the linear integral, which we wish to prove to be equal to the given surface integral, the integration must be extended over the whole boundary in the positive direction as just defined.

We

shall write, then, the integral

J in the form

and we can then integrate in the first part as to # and in the second part as to y. For this purpose we divide the region into elementary strips, which are formed by straight lines lying infinitely near to one another and, for the first integral, in case there are branchrunning parallel to the o>axis ;

points,

we take

The whole region

is

draw such a

line through each of them. thus divided into infinitely narrow trape-

care to

THEORY OF FUNCTIONS.

86

zoid-like strips. In Fig. 22, for instance, in a surface consisting of two sheets and bounded by a closed line -which makes

a circuit twice round a branchpoint, several such trapezoidlike pieces are represented,

the lines running in the second sheet being dotted. If

we now these

select

some one of

elementary

strips,

be-

longing to an arbitrary value of y (i.e., in case the surface consists of several sheets, all

Fra

those elementary strips lying one directly below another in

23>

the different sheets which be-

long to the

same value of

acquired by

the function

mentary

strips

y),

and

if

the values

at those places

in the direction of the positive

(i.e.,

we denote

where the elecut the boundary, counting from left to right

Q

cc-axis),

at the points of

entrance by

and

at the points of exit

by

G',

Q", Q'",

-,

then (Fig. 23)

therefore,

1 It

must be noted that

this equation

remains true, even when

^2 6x

becomes extends,

infinite or discontinuous at if

Q

suffer

some place over which the integration

no interruption of continuity at

this place.

If,

namely, f(x) be a function of the real variable x, which for x = a is continuous, while its derivative, /'(x), is for the same value discontinuous,

INTEGRALS WITH COMPLEX VARIABLES.

87

In the integrals on the right y passes through all values from the least to the greatest therefore dy is always to be taken positively. But if the projections on the 7/-axis of the elementary arcs which have been cut out from the boundary by the elementary strips be designated in the same sequence ;

as above, at the places of entrance

and at the places of

exit

by

by

V*-, we assume on If,

a

lie

the

we

both, sides of a

two values

and x&

Xh

infinitely

near to

a,

then, in the integral

between the

same, while can put

limits X Q

f (x) f

is

and

o&i, and if f(x) remain continuous between discontinuous only at the place x a, then

wherein the limit has reference to the coincidence of x^ and %% with a. Now since f(x) is continuous from XQ to xn and from xk to i, it follows that S*>r

x)dx

is continuous at the place /(XA ) and /(a*) become equal, or

Since /(x) limit,

= lim [/(%)

/(aso) 4-

cc

= a,

/(^i)

/(**)]

therefore, in passing to the

therefore, notwithstanding the discontinuity of of the integral, we have

f(x) between

the limits

This case deserves notice here, since it will be shown later that the derivatives of continuous functions can become infinite at branch-points

THEOEY OF FUNCTIONS.

88

and

if regard be paid to the positive (Fig 23), then

dy

=

dyl =z

dy2

=

boundary direction

dys

=

therefore,

In

all

these integrals ?/ changes in the sense of the positive therefore they all reduce to a single one,

boundary-direction

;

and we have

the latter integral be extended along the entire boundary in the positive direction. In the same manner the second integral if

can be treated.

Here the region

is

divided into elementary

strips by straight lines running parallel to the y-axis, and, as before, such a line is drawn through each branch-point. If,

therefore, the values

which the function

P has

at the places

INTEGRALS WITH COMPLEX VARIABLES.

89

where an elementary strip cuts the boundary be designated, in order from below upward (i.e., in the direction of the positive y-axis), at the places of

entrance by

P&

PI, P.2)

and

at the places of exit

,

by

pu Pi JT y

}

-pin JT ,

',

then again

and therein dx

is positive.

dak dx2 dxS)

But

if

and dx dx u f

**,

,

}

,

dx'",

designate the projections of the elementary arcs which are cut out by the elementary strips, then, considering the positive direction of the boundary,

= + dx = + dx = + dx = - _ dx' = - dx" = - dx' =

dx

l

2

3

and therefore

--

(*

in which the integral is to be extended in the positive direction round the entire boundary. Combining the two integrals, it follows, as

was to be proved, that

the linear integral to be taken round the entire boundary in the positive direction.

This proposition, which is hereby proved for the real funcP and Q, can at once be extended to the case when and

P

tions

Q

are complex functions of the real variables x

put

P= P< + iP",

Q=

Q'

+ t'Q",

and

y.

If

we

THEORY

90 wherein

Q Qn f

P',

P",

,

Off

FUNCTIONS.

are real functions of x

\Sx

By

and

y,

then

By

If the proposition be applied to the right side of the equation,

we

get

We

have assumed until now that, within the region under consideration, there are no branch-points or other points at which P and Q are discontinuous. RTow, in order to include within our considerations also those regions in which this is the case, it is only necessary to enclose, and thereby exclude, such points by arbitrary small closed lines, these new lines then forming part of the boundary of the region. 18. From the preceding proposition, follows immediately the following :

(i.) I

IfPdx-t- Qdy be a complete

(Pdx if

whole boundary of a region

4~ Qdy), extended over the

within which

For,

differential, then the integral,

P and

Q

Pdx + Qdy be

are finite and continuous, is equal to zero.

a complete differential,

SP = SQ

3

By

x

and therefore

all the elements of the surface integral, which is equal to the linear integral, disappear, and accordingly this, as well as that, is equal to zero.

If

now

w=f(z)

be a function of a complex variable z

.-.^- 3

= x + iy,

then

[

5.

(1)]

INTEGRALS WITH COMPLEX VARIABLES. wdx

therefore is

+ iwdy,

w (dx

i.e.,

+ idy)

9

or

91

wdz

a complete differential, and hence (ii.)

f(z) dz

j

= 0,

if this integral be extended

boundary of a region within which f(z)

From

this follows further

:

describe between the points a

^s finite

round

and

continuous.

If the variable z be

and

the whole

made

to

two different paths acb forming together a closed ^ b

and adb (Fig. 24), which in itself alone is the complete boundary of a region, and if f(z) be finite and con-

line

tinuous within this region, then, for the integral extended round the closed line, we have

In order to designate briefly an integral taken along a definite path, we shall choose the letter J"and add to it the path of integration in parenthesis,

so

that,

for instance,

the integral

FIG. 24.

if(z)dz, taken along the path acb, will be designated J(acb).

The

last equation

J(acbda)

But

(

16)

= 0.

= J(acV) + J(bda) J(bda) = J(adb) = that J(acb) J(adfy.

J(acbda)

and .it

by

can be written

;

follows, therefore,

(iii.)

The

integral

J

f(z)dz, therefore, has always the same

value along two different paths joining the same points, if the two paths taken together be the boundary of a region in which f(z) is finite and continuous. If

we have a connected region within which

f(z) remains

and continuous, of such, a nature that every closed line described in it forms by itself alone a complete boundary of a

finite

THEORY OF FUNCTIONS.

92

part of the region, then the integral

I

f(z) dz has, along all

paths between the two points, the same value. Let the lower limit be constant then within such a part of the region the integral is a uniform function of the upper limit, and if F(z) ;

denote a function the derivative of which

is

/(#), then within

this region

since in this case the value of the integral is independent of

the path of integration. The great importance of those surfaces, in which each closed line forms by itself alone the com-

boundary of a region, becomes here quite evident. called surfaces of such a character simply connected surfaces. Such is, for instance, the surface within a circle. If /(V) be continuous everywhere in such a surface, then, as has been noted,

plete

Eiemann has

"

Voo